A235590 Sum of parts of the form 10...0 with nonnegative number of zeros in binary representation of c-squarefree numbers (A233564) as the corresponding powers of 2.

1, 2, 4, 3, 3, 8, 5, 5, 16, 9, 6, 6, 9, 32, 17, 10, 7, 7, 10, 7, 7, 17, 7, 7, 64, 33, 18, 12, 11, 11, 12, 18, 11, 11, 33, 11, 11, 128, 65, 34, 20, 19, 19, 13, 13, 20, 13, 13, 34, 19, 19, 65, 19, 13, 13, 19, 256, 129, 66, 36, 35

Offset: 1

Author

Vladimir Shevelev, Jan 12 2014

Keywords

• nonn
• base

Comments

Subsequence of A162439.

The number of times of appearances of number k in the sequence is the number of compositions of k into distinct powers of 2, i.e., it is A000120(k)!

Examples

			Let n=17, A233564(17)=37. In binary a concatenation of parts of the form 10...0 which gives 37 is (100)(10)(1). Thus a(17)= 4+2+1 = 7.
		

Crossrefs

Cf. A233564, A162439, A000120.

Programs

Mathematica

bitPatt[n_]:=bitPatt[n]=Split[IntegerDigits[n,2],#2==0&]; Map[Plus@@(Map[FromDigits[#,2]&,bitPatt[#]])&,Select[Range[300],#==DeleteDuplicates[#]&[bitPatt[#]]&]] (* Peter J. C. Moses, Jan 15 2014 *)

Formula

Let, for k_1>k_2>...>k_r, A233564(n) = 2^k_1 + 2^k_2 +...+ 2^k_r. Then a(n) = 2^(k_1-k_2-1) + 2^(k_2-k_3-1) + 2^(k_(r-1)-k_r-1) + 2^k_r.